#coeff. restituzione=vel dopo impatto/vel prima impatto
#IC 0.95

dati<-
  c(
0.8411,
0.8125,
0.8532,
0.7983,
0.8282,
0.8191,
0.8750,
0.8483,
0.8042,
0.8359,
0.8182,
0.8580,
0.8276,
0.8730,
0.8660
)
#tabella di lavoro
#media<-sum(dati)/n
table<-cbind(dati,dati^2)
#,(dati-media),(dati-media)^2)
colnames(table)<-c("dati","dati^2")
#,"scarti","scarti^2")
table
sommetable<-colSums(table)
sommetable
TABLE<-rbind(table,sommetable)
rownames(TABLE)<-c(1:15,"somme")
TABLE
n<-15
med<-sommetable[1]/n

S2<-(sommetable[2]/n-med^2 )*n/(n-1)
#oppure i comandi
#mean(dati)
#var(dati)


#sigma noto#alpha=0.95##############################
varianza<-0.0005
media<-mean(dati)#sum(dati)/n
n<-length(dati)
z0975<-qnorm(1-0.05/2)
L<-media-z0975*sqrt(varianza)/sqrt(n)
U<-media+z0975*sqrt(varianza)/sqrt(n)
IC095<-c(L,U)
IC095

#numerosità necessaria per dimezzare ampiezza intervallo
d<-(U-L)/2
nn<-(   2*z0975*sqrt(varianza)/d   )^2
#per ridurla a un quarto
d<-(U-L)/4
nnnn<-(   2*z0975*sqrt(varianza)/ d )^2


#sigma non noto############################################
varianza<-var(dati)
media<-mean(dati)#sum(dati)/n
n<-length(dati)
t0975<-qt(1-0.05/2,n-1)
L<-media-t0975*sqrt(varianza)/sqrt(n)
U<-media+t0975*sqrt(varianza)/sqrt(n)
IC095<-c(L,U)
IC095

#######intervalli unilaterali [L Inf) )


alpha<-1-(0.95)
zsx<-qt(alpha,n-1)
L<-media+zsx*sqrt(varianza)/sqrt(n)
c(L,Inf)
#ESERCIZIO  intervallo (-Inf U]
U<-media+qt(0.95,n-1)*sqrt(varianza)/sqrt(n)
c(-Inf,U)
#IC per varianza alpha 0.95
c025<-qchisq(0.025,n-1)
c0975<-qchisq(0.975,n-1)
l<-(n-1)*varianza/c0975
u<-(n-1)*varianza/c025
ic<-c(l,u)
ic




########################TEST DI IPOTESI
#########test valore atteso uo=0.84 alpha 0.05 varianza non nota
varianza<-var(dati)
media<-mean(dati)
n<-length(dati)
test<-(media-0.84)/sqrt(varianza)*sqrt(n)
#alternativa dx
c<-qt(0.95,n-1)
#decisione
test>=c
P<-1-pt(test,n-1)
P<=0.05
#alternativa sx
c<-qt(0.05,n-1)
#decisione
test<=c
P<-pt(test,n-1)
P<=0.05
#alternativa bil
c<-qt(0.975,n-1)
#decisione
test<=-c|test>=c
#########test valore atteso uo=0.84 alpha 0.05 varianza  nota
varianza<-0.0005
test<-(media-0.84)/sqrt(varianza)*sqrt(n)
#alternativa dx
c<-qnorm(0.95)
#decisione
test>=c
#alternativa sx
c<-qnorm(0.05)
#decisione
test<=c

#alternativa bil
c<-qnorm(0.975)
#decisione
test<=-c|test>=c


######################################test varianza sigma20=0.002
n<-length(dati)
varianza<-var(dati)
test<-(n-1)*varianza/0.002
#alternativa DX alpha=0.05
c<-qchisq(0.95,n-1)
test>=c
pvalue<-1-pchisq(test,n-1)
pvalue
#alternativa SX alpha=0.05
c<-qchisq(0.05,n-1)
test<=c
pvalue<-pchisq(test,n-1)
pvalue
#alternativa bil
c1<-qchisq(0.025,n-1)
c2<-qchisq(0.975,n-1)
test<=c1|test>=c2
pvaluesx<-pchisq(test,n-1)
  pvaluedx<-(1-pchisq(test,n-1))
pvalue<-2*min(pvaluesx,pvaluedx)
pvalue
#######################IC###########################
#lenticchie spa peso confezioni di lenticchie normale sigma=12.6
#n=32 ,media campionaria 97.3,1-alpha=0.99 IC per mu
m<-97.3
n<-32
sigma<-12.6
p<-1-(1-0.99)/2 #1-alpha/2
z<-qnorm(p)
L<-m-z*sigma/sqrt(n)
U<-m+z*sigma/sqrt(n)
c(L,m,U)
#illimitato sx
L<--Inf
U<-m+qnorm(0.99)*sigma/sqrt(n)
c(L,U)
#illimitatodx
U<-Inf
L<-m+qnorm(1-0.99)*sigma/sqrt(n)
c(L,U)
#p=0.001 prob di sovrastima
z_l<-qnorm(0.001)
L<-m+z_l*sigma/sqrt(n)
#p<-pnorm(z_l)
z_u<-qnorm(0.99+0.001)
U<-m+z_u*sigma/sqrt(n)
c(L,U)
#####1-p=1-0.999=0.001 prob sottostima

z_u<-qnorm(0.999)
z_l<-qnorm(1-0.99-(1-p))
U<-m+z_u*sigma/sqrt(n)
L<-m+z_l*sigma/sqrt(n)
c(L,U)

######################################################################
#n=40 impianti irrigazione simili   media consumi annui 750 litri e S=31.3
#1-alpha=0.99 IC per conumo atteso annuo
n<-40
m<-750
s<-31.3
p<-1-(1-0.99)/2
z<-qt(p,n-1)
L<-m-z*s/sqrt(n)
U<-m+z*s/sqrt(n)
c(L,m,U)
#per varianza normale
################################################################################
#7 pezzi su 90 risultano difettosi, 1-alpha=0.95, IC pe rprobabilitàpezzo difettoso
P<-7/90
n<-90
p<-1-(1-0.95)/2
z<-qnorm(p)
L<- P-z*sqrt( P*(1-P) )/sqrt(n)
U<- P+z*sqrt( P*(1-P) )/sqrt(n)
c(L,P,U)
##################################################################################
#Il peso sfere prodotte è una vc normale. In un campione di 41 sfere S^2=15.1
#1-alpha=0.95 IC
n<-41
S2<-15.1
p<-(1-0.95)/2
P<-1-(1-0.95)/2
z_p<-qchisq(p,n-1)
z_P<-qchisq(P,n-1)
L<-(n-1)*S2/z_P
U<-(n-1)*S2/z_p
c(L,S2,U)
########################################################################################
#si è interessti a diametro sfere prodotte. Diametri vc normali con sigma=3.1
#si vuole IC per valore atteso diametri 1-alpha=0.95 d=0.3 Cosa deve essere n
sigma<-3.1
d<-0.3
p<-1-(1-0.95)/2
z<-qnorm(p)
n<-(2*z*sigma/d)^2
ceiling(n)
#########################################################################################
#intervallo confidenza probabilità pezzi difettosi 1-alpha=0.95 d=0.06
d<-0.06
p<-1-(1-0.95)/2
z<-qnorm(p)
n<-(2*z*1/4/d)^2
ceiling(n)



#######################TEST###########################
#media campionaria pesi prodotto  numerosità n=11
#sigma=1.2 u0=7.3 verso tre alternative alpha=0.05
dati<-c(7.8,6.6,6.5,7.4,7.3,7,6.4,7.1,6.7,7.6,6.8)

alpha<-0.05
m<-mean(dati)
n<-length(dati)
sigma<-1.2
u0<-7.3
#altern dx
test<-(m-u0)/sigma*sqrt(11)
zdx<-qnorm(1-alpha)
#regola
test>=zdx
pvalue<-1-pnorm(test)
pvalue<=alpha
#altern sx
zsx<-qnorm(alpha)
#regola
test<=zsx
pvalue<-pnorm(test)
pvalue<=alpha
#alternbil
zbil<-qnorm(1-alpha/2)
#regola
test<=-zbil|test>=zbil
pvalue<-2*(1-pnorm(abs(test)))
pvalue<=alpha
####come sopra ma varianza non nota
u0<-7.3
alpha=0.05

dati<-c(7.8,6.6,6.5,7.4,7.3,7,6.4,7.1,6.7,7.6,6.8)
m<-mean(dati)
varianza<-var(dati)
n<-length(dati)
u0<-7.3
test<-(m-u0)/sqrt(varianza)*sqrt(11)
zdx<-qt(1-alpha,n-1)
#regola
test>=zdx
pvalue<-1-pt(test,n-1)
pvalue<=alpha
#altern sx
zsx<-qt(alpha,n-1)
#regola
test<=zsx
pvalue<-pt(test,n-1)
pvalue<=alpha
#alternbil
zbil<-qt(1-alpha/2,n-1)
#regola
test<=-zbil|test>=zbil
pvalue<-2*(1-pt(abs(test),n-1))
pvalue<=alpha
###########################################################
#dati esercizio sopra testare sigma2=1.44 alpha=0.05 alt bilat
test=(n-1)*var(dati)/1.44
chi1<-qchisq(0.05/2,n-1)
chi2<-qchisq(1-0.05/2,n-1)
test<=chi1|test>=chi2
########test su proporzione test per probabilità
#7 pezzi su 90 risultano difettosi, alpha=0.01, test prprobabilitàpezzo difettoso
#p0=0.01 alternativa dx
#dati
P<-7/90
n<-90
p0<-0.05
#

test<-(P-p0)/sqrt(p0*(1-p0))*sqrt(n)
z099<-qnorm(0.99)
test>=z099
(1-pnorm(test))<=0.01
#############################potenza
#media campionaria pesi prodotto  numerosità n=11
#sigma=1.2 u0=7.3  alpha=0.01 unilat dx potenza verso ua=8
u0<-7.3
ua<-8
sigma<-1.2
n<-11
la<-(ua-u0)/sigma
z099<-qnorm(0.99)
zpot<-z099-la*sqrt(n)
pot<-1-pnorm(zpot)
#n per pot=0.90
zstar<-qnorm(1-0.90)
n<-(    (z099-zstar)/la  )^2
ceiling(n)

#media campionaria pesi prodotto  numerosità n=11
#sigma=1.2 u0=7.3  alpha=0.01 unilat sx potenza verso ua=6.6
u0<-7.3
ua<-6.6
sigma<-1.2
n<-11
la<-(ua-u0)/sigma
z001<-qnorm(0.01)
zpot<-z001-la*sqrt(n)
pot<-pnorm(zpot)
#n per pot=0.90
zstar<-qnorm(0.9)
n<-(    (z001-zstar)/la  )^2
ceiling(n)

###########################
#############################potenza
#media campionaria pesi prodotto  numerosità n=11
#sigma non noto la=0.58  alpha=0.01 unilat dx potenza verso ua=8
n<-39
la<-0.58
z099<-qt(0.99,n-1,0)
alpha<-1-pt(z099,n-1,0)
nc<-la*sqrt(n)
pot<-1-pt(z099,n-1,nc)
pot
#media campionaria pesi prodotto  numerosità n=11
#sigma non noto la=-0.58  alpha=0.01 unilat sx potenza verso ua=8
n<-39
la<--0.58
z001<-qt(0.01,n-1,0)
alpha<-pt(z001,n-1,0)
nc<-la*sqrt(n)
pot<-pt(z001,n-1,nc)
pot

############tet 2 campioni##########################################################################
#library(xtable)
x<-c(9.89 , 10.05 , 9.01 , 9 , 9.9 , 6.84 , 9.95 ,8.8 , 8.8 , 7.9 , 9 , 11)
y<-c( 10.33 ,8.95 , 10.5 ,7.7 , 9.6 , 8 , 8.6, 9.8 , 9 , 8.8 , 9.2 ,10)

t.test(x,y,var.equal=TRUE,conf.level=0.99,alterntive="two.sided")
t.test(x,mu=8,alternative="greater",conf.level=0.99)

#calcoli diretti
mx<-mean(x)
sx<-var(x)
my<-mean(y)
sy<-var(y)
#ancora più elementary
#mx2<-(  sum(x^2)/12-(sum(x)/12)^2   )*12/11

#test due campioni
ss<-(11*sx+11*sy)/22
stxy<-(mx-my)/sqrt(ss*(1/12+1/12))
stxy
p<-(1-0.99)/2
z099<-qt(p,22)
stxy<=-z099|stxy>=z099
PBIL<-2*(1-pt(abs(stxy),22))
PBIL
PBIL<=1-0.99
#se sigma2^2=1 noto
sigma2<-1
stxy<-(mx-my)/sqrt(sigma2*(1/12+1/12))
stxy
p<-1-(1-0.99)/2
z099<-qnorm(p)
stxy<=-z099|stxy>=z099
PBIL<-2*(1-pt(abs(stxy),22))
PBIL
PBIL<=1-0.99


F<-sx/sy
p025<-qf(0.025,11,11)
p975<-qf(0.975,11,11)
F<=p025|F>=p975





#library(xtable)
#colnames(working.table)<-c("x","y","x^2","y^2","x*y")
#edit(working.table)
#workreg<-xtable(TABLE,caption="IC TEST-tabella di lavoro",digits=4)
#print(workreg,file="IC.tex")